Note that taking the average with respect to the distribution of the covariates \(\mathbf {X}_{ij}\)  gives us the overall average treatment effect on the treated (ATT), i.e. \({\rm ATT}(t) = \mathbb {E}\left[ \textrm {CATT}(\mathbf {X}_{ij}, t)\right]\) . For both estimands (and contrary to the Data Challenge), the expectation is taken over a hypothetical super-population of practices and beneficiaries at each time \(t\)  in order to better align with the Bayesian paradigm we pursue in the paperThe super-population we have in mind is based on “model-based” sampling [1]} where for each time \(t\) , the entire collection of random variables \((Y_{11t}(1), Y_{11t}(0), Z_1, \mathbf {X}_{11},\ldots ,Y_{n_{jt}Jt}(1), Y_{n_{jt}Jt}(0), Z_J, \mathbf {X}_{n_{jt}J})\)  follows some distribution \(F\)  and the expectation is over the distribution \(F\) . For example, suppose at each time \(t\) , we have \((Y_{ijt}(1), Y_{ijt}(0), Z_j, \mathbf {X}_{ij})\stackrel{\text{i.i.d.}}{\sim } G\)  where \(G\)  can be a parametric distribution or a mixture of parametric distributions, say a mixture of \(J\)  parametric densities where each density represents one of the \(J\)  practices. Then \(F = \prod _{ij} G\)  and the expectation \(\mathbb {E}\left[Y_{ijt}(1)-Y_{ijt}(0) \vert Z_j = 1, \mathbf {X}_{ij} = \mathbf {x}\right]\) , which is over \(F\) , is equivalent to the expectation over \(G\) . Historically, model-based sampling has been associated with Bayesian paradigms as it allows investigators to encode subject-matter expertise by imposing constraints on \(F\) ; see, e.g., [2]}, [1]}, [4]}, and and [5]} for further discussions. Under appropriate conditions, an estimator (or a test) of the super-population estimand is also reasonable for estimating the finite-sample analog of the estimand; see [6]} for one example and [7]}..
